Weight painting problem : when I move the head of my character,the cloak also move but it shouldn't
If you want to simulate the cloak with the cloth simulator, you'll have to remodel it. This cloak was made for a still model, it was not meant to be simulated. Two points to keep in mind :


A cloth to be simulated should have a non distorted starting shape, since the simulation will perform the distortions on it. If you leave the base mesh distorted, the surface will be completely uneven. It will look like some parts of the fabric are stretched.
a cloth simulation only works if the cloth is single-sided. Your cloak has 2 sides. If you perform a cloth simulation on it, the simulator will still consider every face a piece of fabric. So you're basically telling it this mesh is a double cloak (except both sides are joined at their boundaries). To give thickness to a single-sided simulated cloth you have to use a solidify modifier after the cloth modifier.


As for attaching parts to your character, you have to combine pinning and hooking :


In pose mode, select the bone you want your pinned cloth to follow
Shift select your cloth
Tab to enter edit mode
select the vertices of your cloth you want to attach
create a new vertex group and assign the selected vertices
In the vertex operators (Ctrl+V), go in Hooks > "Hook to selected object bone"
Move the hook modifier on top of the cloth modifier
In the cloth properties, enable Pinning and select the vertex group you just created.


Here's a simple example of the result :



And here's the file used for this gif :